Why do countries build domestic airport and international airport apart?

One idea that I had during writing this answer:

Of course you end up having domestic flights from the "international" airport, too. You can charge more for those flights, while subsidizing flights from the "domestic" airports. This way you can have cheap flights around the country, while making international travellers to the rest of the country foot a large bill.

Sounds like something that looked like a good idea in the 60s or so, but now is actively harmful because it starves inner regions for tourists and business travellers, incentivizes everyone to move to the capital/largest city, which is something most countries try to avoid these days.

If your city has two airports, typically one of them will be "better" than the other - e.g. more convenient to the city center, more modern facilities, etc. So without any restrictions, traffic would tend to gravitate toward the better one. Then you don't really have the benefit of two airports - the better one is overcrowded, and the worse one sits underutilized. Traffic isn't divided between them in an efficient manner.

So you may want to have a rule that forces both airports to be used. And one simple such rule is to declare that one airport is only (or mainly) for international traffic, and the other for domestic. This has the benefit that travelers can know immediately which one they should use.

Of course it has drawbacks too, as you have mentioned.

As far as having multiple separate airports within the same city, I watched a video on why London has as many as six different airports serving the city. Basically, it was because planes got bigger and noisier, and so it was necessary to demolish an older airport to build a newer one with bigger runways and newer facilities. DJClayworth's answer explains this in better detail.

As far as operating flights within a separate terminal in the same airport: one modern reason for doing so is operating cost for the airline. I can think of two modern examples:

• Austin-Bergstrom International Airport (Austin, Texas, USA) recently opened a new "South Terminal" that is completely separate from the main terminal. The South Terminal has no jetbridges; people board planes by stairs. Ultra-low-fare carriers such as Allegiant Air operate from this terminal, and similar carriers such as Frontier Airlines and Spirit Airlines have plans to move their operations there.

• Chhatrapati Shivaji International Airport (Mumbai, India) used to have all domestic flights run from Terminal 1, while all international flights ran from Terminal 2 (hence the local lingo "domestic airport" and "international airport"). Later on, Terminal 2 was rebuilt and replaced with a much-expanded terminal, which is capable of operating both domestic and international flights. The original plan was to demolish Terminal 1 and operate all flights in a single terminal, but low-fare airlines such as IndiGo still preferred to operate out of Terminal 1, which is why it still exists (though parts of it have been decommissioned).

In both the above cases, both terminals can't be accessed from the same access road; you need to drive outside the airport property to access one from the other. Both of the low-cost terminals (Austin's South Terminal and Mumbai's Terminal 1) have a significantly lower operating cost for the airline than the main terminals. For instance, not having jetbridges costs the airline less than having jetbridges (though Mumbai's Terminal 1 does have some jetbridges; very few flights use them). The savings are often passed to the consumer through lower fares.

Most of the time, these low-cost terminals lack the necessary immigration and customs facilities needed to support international flights. While the U.S. does not have exit immigration, it still requires incoming international passengers to pass through entry immigration and customs. (In some cases, these checks are done at the origin airport; this is why U.S. airports that ordinarily only have domestic flights can also have flights to preclearance destinations.)